"What the Tortoise Said to Achilles", written by Lewis Carroll in 1895 for the philosophical journal Mind, is a brief dialogue which problematises the foundations of logic. The title alludes to one of Zeno's paradoxes of motion, in which Achilles could never overtake the tortoise in a race. In Carroll's dialogue, the tortoise challenges Achilles to use the force of logic to make him accept the conclusion of a simple deductive argument. Ultimately, Achilles fails, because the clever tortoise leads him into an infinite regression.

B: "The two sides of this triangle are things that are equal to the same"

Therefore Z: "The two sides of this triangle are equal to each other"

The Tortoise asks Achilles whether the conclusion logically follows from the premises, and Achilles grants that it obviously does. The Tortoise then asks Achilles whether there might be a reader of Euclid who grants that the argument is logically valid, as a sequence, while denying that A and B are true. Achilles accepts that such a reader might exist, and that he would hold that ifA and B are true, thenZ must be true, while not yet accepting that A and Bare true. (A reader who denies the premises.)

The Tortoise then asks Achilles whether a second kind of reader might exist, who accepts that A and Bare true, but who does not yet accept the principle that ifA and B are both true, thenZ must be true. Achilles grants the Tortoise that this second kind of reader might also exist. The Tortoise, then, asks Achilles to treat the Tortoise as a reader of this second kind. Achilles must now logically compel the Tortoise to accept that Z must be true. (The tortoise is a reader who denies the argument itself; the syllogism's conclusion, structure, or validity.)

After writing down A, B, and Z in his notebook, Achilles asks the Tortoise to accept the hypothetical:

C: "If A and B are true, Z must be true"

The Tortoise agrees to accept C, if Achilles will write down what it has to accept in his notebook, making the new argument:

A: "Things that are equal to the same are equal to each other"

B: "The two sides of this triangle are things that are equal to the same"

C: "If A and B are true, Z must be true"

Therefore Z: "The two sides of this triangle are equal to each other"

But now that the Tortoise accepts premise C, it still refuses to accept the expanded argument. When Achilles demands that "If you accept A and B and C, you must accept Z," the Tortoise remarks that that's another hypothetical proposition, and suggests even if it accepts C, it could still fail to conclude Z if it did not see the truth of:

D: "If A and B and C are true, Z must be true"

The Tortoise continues to accept each hypothetical premise once Achilles writes it down, but denies that the conclusion necessarily follows, since each time it denies the hypothetical that if all the premises written down so far are true, Z must be true:

"And at last we've got to the end of this ideal racecourse! Now that you accept A and B and C and D, of course you accept Z."

"Do I?" said the Tortoise innocently. "Let's make that quite clear. I accept A and B and C and D. Suppose I still refused to accept Z?"

"Then Logic would take you by the throat, and force you to do it!" Achilles triumphantly replied. "Logic would tell you, 'You can't help yourself. Now that you've accepted A and B and C and D, you must accept Z!' So you've no choice, you see."

"Whatever Logic is good enough to tell me is worth writing down," said the Tortoise. "So enter it in your notebook, please. We will call it

(E) If A and B and C and D are true, Z must be true.

Until I've granted that, of course I needn't grant Z. So it's quite a necessary step, you see?"

"I see," said Achilles; and there was a touch of sadness in his tone.

Thus, the list of premises continues to grow without end, leaving the argument always in the form:

(1): "Things that are equal to the same are equal to each other"

(2): "The two sides of this triangle are things that are equal to the same"

(3): (1) and (2) ⇒ (Z)

(4): (1) and (2) and (3) ⇒ (Z)

...

(n): (1) and (2) and (3) and (4) and ... and (n − 1) ⇒ (Z)

Therefore (Z): "The two sides of this triangle are equal to each other"

At each step, the Tortoise argues that even though he accepts all the premises that have been written down, there is some further premise (that if all of (1)–(n) are true, then (Z) must be true) that it still needs to accept before it is compelled to accept that (Z) is true.

Lewis Carroll was showing that there is a regress problem that arises from modus ponens deductions.

The regress problem arises because, in order to explain the logical principle, we have to propose a prior principle. And, once we explain that principle, we have to introduce another principle to explain that principle. Thus, if the causal chain is to continue, we are to fall into infinite regress. However, if we introduce a formal system where modus ponens is simply an axiom, then we are to abide by it simply, because it is so. For example, in a chess game there are particular rules, and the rules simply go without question. As players of the chess game, we are to simply follow the rules. Likewise, if we are engaging in a formal system of logic, then we are to simply follow the rules without question. Hence, introducing the formal system of logic stops the infinite regression—that is, because the regress would stop at the axioms or rules, per se, of the given game, system, etc. Though, it does also state that there are problems with this as well, because, within the system, no proposition or variable carries with it any semantic content. So, the moment you add to any proposition or variable semantic content, the problem arises again, because the propositions and variables with semantic content run outside the system. Thus, if the solution is to be said to work, then it is to be said to work solely within the given formal system, and not otherwise.

Some logicians (Kenneth Ross, Charles Wright) draw a firm distinction between the conditional connective (the syntactic sign "→"), and the implication relation (the formal object denoted by the double arrow symbol "⇒"). These logicians use the phrase not p or q for the conditional connective and the term implies for the implication relation. Some explain the difference by saying that the conditional is the contemplated relation while the implication is the asserted relation.[citation needed] In most fields of mathematics, it is treated as a variation in the usage of the single sign "⇒," not requiring two separate signs. Not all of those who use the sign "→" for the conditional connective regard it as a sign that denotes any kind of object, but treat it as a so-called syncategorematic sign, that is, a sign with a purely syntactic function. For the sake of clarity and simplicity in the present introduction, it is convenient to use the two-sign notation, but allow the sign "→" to denote the boolean function that is associated with the truth table of the material conditional.

These considerations result in the following scheme of notation.

The paradox ceases to exist the moment we replace informal logic with propositional logic[citation needed]. The Tortoise and Achilles don't agree on any definition of logical implication. In propositional logic the logical implication is defined as follows:

Hence de modus ponens [P ∧ (P → Q)] ⇒ Q, is a valid logical implication according to the definition of logical implication just stated. There is no need to recurse since the logical implication can be translated into symbols, and propositional operators such as →. Demonstrating the logical implication simply translates into verifying that the compound truth table is producing a tautology.

Several philosophers have tried to resolve Carroll's paradox. Bertrand Russell discussed the paradox briefly in § 38 of The Principles of Mathematics (1903), distinguishing between implication (associated with the form "if p, then q"), which he held to be a relation between unasserted propositions, and inference (associated with the form "p, therefore q"), which he held to be a relation between asserted propositions; having made this distinction, Russell could deny that the Tortoise's attempt to treat inferringZ from A and B is equivalent to, or dependent on, agreeing to the hypothetical "If A and B are true, then Z is true."

The Wittgensteinian philosopher Peter Winch discussed the paradox in The Idea of a Social Science and its Relation to Philosophy (1958), where he argued that the paradox showed that "the actual process of drawing an inference, which is after all at the heart of logic, is something which cannot be represented as a logical formula ... Learning to infer is not just a matter of being taught about explicit logical relations between propositions; it is learning to do something" (p. 57). Winch goes on to suggest that the moral of the dialogue is a particular case of a general lesson, to the effect that the proper application of rules governing a form of human activity cannot itself be summed up with a set of further rules, and so that "a form of human activity can never be summed up in a set of explicit precepts" (p. 53).

Hofstadter, Douglas. Gödel, Escher, Bach: an Eternal Golden Braid. See the second dialogue, entitled "Two-Part Invention". Hofstadter appropriated the characters of Achilles and the Tortoise for other, original, dialogues in the book which alternate contrapuntally with prose chapters. Hofstadter's Tortoise is of the male sex, though the Tortoise's sex is never specified by Carroll. The French translation of the book rendered the Tortoise's name as "Madame Tortue".